Question

a boat is heading towards a lighthouse, where madeline is watching from a vertical distance of 140 feet above the water. madeline measures an angle of depression to the boat at point $a$ to be $5^{\circ}$. at some later time, madeline takes another measurement and finds the angle of depression to the boat now at point $b$ to be $50^{\circ}$. find the distance from point $a$ to point $b$. round your answer to the nearest tenth of a foot if necessary.

Explanation:

Step1: Relate angles of depression to elevation

The angle of depression equals the angle of elevation from the boat to Madeline. So, $\angle BAL = 5^\circ$, $\angle BLL' = 50^\circ$ (where $L$ is the base of the lighthouse). We use right triangles $\triangle ALL'$ and $\triangle BLL'$ with height $140$ ft.

Step2: Calculate distance from A to L

Use $\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. For point A:
$\tan(5^\circ) = \frac{140}{AL}$
$AL = \frac{140}{\tan(5^\circ)}$
$AL \approx \frac{140}{0.0875} \approx 1600.99$ ft

Step3: Calculate distance from B to L

For point B:
$\tan(50^\circ) = \frac{140}{BL}$
$BL = \frac{140}{\tan(50^\circ)}$
$BL \approx \frac{140}{1.1918} \approx 117.47$ ft

Step4: Find distance A to B

Subtract $BL$ from $AL$:
$AB = AL - BL$

Answer:

$1600.99 - 117.47 = 1483.5$ feet